DOMAIN W ALL MOTION IN
NANOWIRES
R.WIESER∗,U.NOWAK and K.D.USADEL
Institute of Physics,
University of Duisburg-Essen,47048Duisburg,Germany
()
Abstract
We investigated the motion of domain walls in ferromagnetic cylin-drical nanowires by solving the Landau-Lifshitz-Gilbert equation nu-
merically for a classical spin model in which energy contributions from
exchange,crystalline anisotropy,dipole-dipole interaction,and a driv-
ing magneticfield are considered.Depending on the diameter either
transverse domain walls or vortex walls are found.The transverse do-
main wall is observed for diameters smaller than the exchange length
of the given system.In this case,the system behaves effectively one
dimensional and the domain wall velocity agrees with a result for one
dimensional walls by Slonczewski.For larger diameter a crossover to
a vortex wall sets in which enhances the domain wall velocity drasti-阎娜的胸围多大
cally.For a vortex wall the domain wall velocity is described by the
Walker formula.
1.INTRODUCTION
Arrays of magnetic nanowires are possible candidates for patterned magnetic storage media(Ross e
t al.(2000);Nielsch et al.(2002)).For these nanowires and also for other future magneto-electronic devices the understanding of domain wall motion and mobility is important for the controlled switching
of the nanostructure.In a recent experiment the velocity of a domain wall in a NiFe/Cu/NiFe trilayer was investigated using the GMR effect(Ono et al. (1999)).The measured velocities were compared with the Landau-Lifshitz formula for domain wall motion(Landau and Lifshitz(1935)).This compar-ison was used to determine the damping constant of the trilayer,a quantity which is usually not known a priori.However,several formulas for the ve-locity of a domain wall can be found in the literature(Landau and Lifshitz (1935);Malozemoffand Slonczewski(1979);Schryer and Walker(1974);Dil-lon(1963);Garanin(1991))which are derived in different limits but all all of them in(quasi)one-dimensional models neglecting the possible influence of non-uniform spin structures within the domain wall.Thus the question arises in how far these formulas are applicable to real three dimensional domain structures.To shed some light onto this problem we numerically investigate the domain wall mobility in nanowires starting from a three dimensional local spin model.
2.MODEL AND SIMULATIONS
你的答案In the following we consider a classical spin model with energy contributions from exchange,crystalline anisotropy,dipole-dipole interaction,and a driving magneticfield.Such a spin model for the description of magnetic nanostruc-tures can be justified following different lines(Nowak(2001)):on the one hand it is the classical limit of a quantum mechanical localized spin model,on the other hand it might be interpreted as the discretized version of a micromagnetic continuum model,where the charge distribution for a single cell of the discretized lattice is approximated by a point dipole.For certain magnetic systems their description in terms of a lattice of magnetic moments may even be based on the mesoscopic structure of the material,especially when a particulate medium is described.
However,our intention is not to describe a particular material but to investigate a general model Hamiltonian which is
H=−J ij S i·S j−µs B· i S i−D e i(S z i)2
−ω i<j3(S i·e ij)(e ij·S j)−S i·S j
sum is the dipolar interaction where w=µ0µ2s/(4πa3)describes the strength of the dipole-dipole interaction.The e ij are unit vectors pointing from lattice site i to j and r ij is the distance between thes
彭佳慧与陈国华e lattice sites in units of the lattice constant a.
The underlying equation of motion for magnetic moments which we con-sider in the following is the Landau-Lifshitz-Gilbert(LLG)equation,
∂S i
(1+α2)µs S
i
× H i(t)+αS i×H i(t) ,(2)
with the gyromagnetic ratioγ=1.76×1011(Ts)−1,the dimensionless damp-ing constantα(after Gilbert),and the internalfield H i(t)=−∂H/∂S i.
In the following we present results from the simulation of cylindrical sys-tems being parallel to the z-axis with a length of256lattice sites and diameter d=8.Our systems are defined on a cubic lattice and consist of dipoles at those lattice sites which reside within the cylinder of given diameter and length.Due to shape as well as crystalline anisotropy the equilibrium mag-netization is aligned with th
e long axis of the system.However,we start the simulation with an abrupt,head-to-head domain wall as initial configura-tion,letting the wall relax until a stable state is reached.The distance of the initial wall position from the end is approximately1/3of the system length. Then we switch on the driving magneticfield B along the easy axis and wait until a stationary state is reached for some time interval in which the velocity v of the wall is constant while the wall is moving through the central part of the wire.We calculate the domain wall velocity from the magnetization versus time data,averaged over a period of time where no influence of the finite system size on the domain wall can be observed,i. e.until the wall approaches the other end of the wire.
3.RESULTS AND DISCUSSION
the saturdaysDepending on the ratioω/J,either transverse domain walls or vortex do-main walls are found.For the static limit(B=0)representative spin con-figurations are shown in Fig.1.The important length scale is the so-called exchange length d ex/a=π
Walker(1974)),
γBa J
v=
α+1/α 2D e.(4)
The difference between both equations is in the dependence on the damp-ing constantα.Both equations are calculated by solving the Landau-Lifshitz-Gilbert equation in different limits(for a review see Malozemoffand Slon-czewski(1979)).Figure2shows the two possible paths for the reversal of a spin.Path(1)is restricted to one plane as is the case for steady state motion,while path(2)is dominated by precession.The case of steady state motion is described by the Walker equation(Eq.3)the more general case with precession is described by the Slonczewski equation(Eq.4).Both equations above were derived for one-dimensional systems,but they can also be used to calculate the velocity in three-dimensional nanowires(Wieser et al.(2004)). For a transverse domain wall we have found a precessional motion.Figure 3shows a moving transverse domain wall for three equidistant times.The precession of the wall follows from the fact that the motion of each single spin consists of precession and relaxation with no restriction to on single plane (path(2)in Figure2).As was shown numerically in this case the domain wall velocity is well described by the Slonczewski equation(Eq.4)(Wieser et al.(2004)).
For a vortex domain wall the situation is different,and the velocity is well described by the Walker formula(Eq.3)(Wieser et al.(2004)).Figure4 shows snapshots of a moving vortex domain wall for thre
e equidistant times. For a qualitative understanding of the spin motion in the case of a vortex domain wall,we note that the motion of the spins within each spin chain, parallel to the wire axis,is restricted to a certain plane passing through the spin positions.For spins at the surface of the cylinder,and in the limit of small drivingfields,these planes are tangential to the surface of the cylinder. The responsible force that keeps the spin motion of each spin chain in this plane is the energetical principle that forms the he combination of exchange and dipolar interaction.
4.SUMMARY
In good agreement with prior work on cylindrical nanowires we have found different wall structures,transverse and vortex domain wall,depending on the diameter of the system compared to the exchange length of the given material.
In both cases the numerically determined wall velocities are in agreement with simple analytical expressions as derived in the one dimensional limit. While for the transverse domain wall the velocity is described by the formula of Slonczewski,the vortex domain wall is described by the formula of Walker. The main difference between both formulas is the dependence on the damping constantα.The reason for this difference is the reversal process during the domain wall motion.In the
case of the vortex domain wall the spin motion is completely within one single plane,while this is not the case for a transverse domain wall where the precession leads to a three-dimensional spin motion. Acknowledgments
We acknowledge the support by the Deutsche Forschungsgemeinschaft(SFB 491).
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